Lecture 03 - Hilbert Spaces

#math

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How the norm affects the shape of the unit sphere on the plane.

Hilbert Space

We know in $R^{n}$ we can take the scalar product of vectors $x, y \in R^{n}$ also known as the inner product which allows us to define orthogonality between vectors.

Definition

Let $V$ be a vector space, a mapping $<, >: V \times V \to R (or C)$ that satisfies the following properties is called an inner product:

$⟨ x, x ⟩ \geq 0$ and $⟨ x, x ⟩ = 0 ⟺ x = 0$
$⟨ x, y ⟩ = \overset{―}{⟨ y, x ⟩}$
$⟨ λ x, y ⟩ = λ ⟨ x, y ⟩$
$⟨ x + y, z ⟩ = ⟨ x, z ⟩ + ⟨ y, z ⟩$

Example

Let $V = C_{b o u n d e d} ([a, b])$ , where $[a, b] \subset R$

\begin{aligned} | | f | |_{\infty} & = \underset{x \in [a, b]}{Max} | f (x) | \\ ⟨ f, g ⟩ & = \int_{a}^{b} f (x) \cdot \overset{―}{g (x)} d x \\ | | f | |_{2} & = {(\int_{a}^{b} | f (x) |^{2})}^{\frac{1}{2}} \end{aligned}

Theorem

(Cauchy-Schwarz Bunyakovski Inequality) For $x, y \in V$ Hilbert space let

| | x | | = \sqrt{⟨ x, x ⟩}

Then we have:

| < x, y > | \leq | | x | | | | y | |

and we have equality if and only if $x = λ y$ .

Proof

$\begin{aligned} 0 < ⟨ x + t y, x + t y ⟩ & =< x, x > + 2 Re (< y, x >) + t^{2} < y, y > \\ \leq | | x | |^{2} + 2 t | ⟨ x, y ⟩ | + t^{2} | | y | |^{2} \end{aligned}$

This is a quadratic equation in $t$ that is non negative thus we can deduce the discriminant is non positive. That is:

4 | ⟨ x, y ⟩ |^{2} - 4 | | x | |^{2} | | y | |^{2} \leq 0

Thus we have proven the inequality.

Definition

A Hilbert space is a complete inner product space.

\begin{aligned} Hilbert Space & \Rightarrow & Banach Spaces & \Rightarrow & Complete metric spaces \\ ⇓ & ⇓ & ⇓ \\ Inner Product Spaces & \Rightarrow & Normed Spaces & \Rightarrow & Metric Spaces \end{aligned}

Proposition

Parallelogram Identity

| | x + y | |^{2} + | | x - y | |^{2} = 2 | | x | |^{2} + 2 | | y | |^{2}

Definition

A subspace of a normed space (or inner product space), we mean a linear subspace with the same norm (or inner product). We write $X \subseteq Y$ .

Example

$C_{b o u n d e d} (X) \subset l_{\infty} (X)$ where $X$ is a metric space.
Any linear subpsace of $R^{n}$ or $C^{n}$ with any of the norms is a subspace.

Proposition

i) Any closed subspace of a Banach/Hilbert space is complete.
ii) Any complete subspace is closed.
iii) The closure of a subspace is a subspace.